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The vacuum F-coefficient records the Frobenius–Schur sign

Statement

Let $a$ and $(v, ev, coev, d_{a}, κ_{a})$ be as in Frobenius–Schur sign in a normalized anyonic vacuum channel. In the multiplicity-free vacuum-normalized gauge of F-symbol as a recoupling matrix,

${(F_{a}^{a a a})}_{1, 1} = \frac{κ_{a}}{d_{a}}, δ_{a} : = \frac{1}{{((F_{a}^{a a a})^{- 1})}_{1, 1}} = κ_{a} d_{a} .$

Facts & Assumptions

Given: A unitary spherical fusion category, a self-dual simple $a$ , the normalized duality datum, and the F-symbol convention.

[F1]

Frobenius–Schur sign in a normalized anyonic vacuum channel: $v^{†} v = {id}_{1}$ , $coev = \sqrt{d_{a}} v$ , and $ev = κ_{a} \sqrt{d_{a}} v^{†}$ , with $κ_{a} = \pm 1$ .

[F2]

F-symbol as a recoupling matrix: in orthonormal tree bases, the coefficient $F_{f, e}$ is defined by $L_{e} \circ α^{- 1} = \sum_{f} F_{f, e} R_{f}$ ; rows index the right-tree channel and columns the left-tree channel.

[F3]

Unitary fusion category and F-symbol unitarity: the chosen trivalent basis maps are orthonormal and the associator F-matrix is unitary.

[F4]

Absolutely simple objects: every simple object has endomorphism space $C id$ , so the overlap and zig-zag scalars are well-defined.

[F5]

Fusion and multifusion categories: $C$ is finite semisimple with simple tensor unit; $a$ is simple by the hypotheses.

Proof

Given: The normalized cap/cup maps, the chosen vacuum-channel bases, and the typed associator between $((a \otimes a) \otimes a)$ and $a \otimes (a \otimes a)$ .

1.1F1F3given

Let $L_{1}$ be the left-tree basis map obtained by applying $v^{†}$ to the first pair and then the unit unitor, and let $R_{1}$ be the right-tree basis map obtained by applying $v^{†}$ to the last pair and then the other unit unitor. Both have norm one because $v^{†} v = {id}_{1}$ and the unit constraints are unitary.

2.1F2F3F4F5step 1.1algebra

By the F-symbol definition, $L_{1} \circ α^{- 1} = \sum_{f} (F_{a}^{a a a})_{f, 1} R_{f}$ . Composing on the right with $R_{1}^{†}$ kills every $f \neq 1$ by orthogonality, so $(L_{1} \circ α^{- 1}) R_{1}^{†} = (F_{a}^{a a a})_{1, 1} {id}_{a}$ . In the chosen basis maps this left side is $(v^{†} \otimes {id}_{a}) α^{- 1} ({id}_{a} \otimes v)$ , with unitors inserted at the two ends.

3.1F1F2F4F5step 2.1algebra

The first duality zig-zag is $(ev \otimes {id}_{a}) α^{- 1} ({id}_{a} \otimes coev) = {id}_{a}$ . Substituting [F1] gives $κ_{a} d_{a} (v^{†} \otimes {id}_{a}) α^{- 1} ({id}_{a} \otimes v) = {id}_{a}$ . Step 2.1 therefore yields $(F_{a}^{a a a})_{1, 1} = 1 / (κ_{a} d_{a}) = κ_{a} / d_{a}$ , since $κ_{a}^{2} = 1$ .

4.1F3step 3.1algebra∎

The entry in step 3.1 is real. Unitarity [F3] gives $((F_{a}^{a a a})^{- 1})_{1, 1} = \overline{(F_{a}^{a a a})_{1, 1}} = κ_{a} / d_{a}$ . Its reciprocal is $δ_{a} = κ_{a} d_{a}$ , proving the normalization with the actual FS sign.

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Sources

Andrei Bernevig and Titus Neupert, Topological Superconductors and Category Theory (standard reference, not scraped)
David Penneys, Unitary Fusion Categories, Math 8800 lecture notes (standard reference, not scraped)